Question

2) Assume the logarithmic transformation f a utility function, for the consumption of two commodities is given by 

ln U = ln4 + 0.5ln X + 0.25lnY

(a) if the price of X is GHS2.50 and that of Y is GHS4.00, calculate the optimal combination for an income of GHS50.00.

b) Determine and interpret the value of the Lagrange multiplier.

Answer 1

Part a

$\ln U =\ln4 + 0.5\ln x + 0.25 \ln y\\ U =4 + e^{0.5\ln x} + e^{0.25 \ln y}=0\\ \frac{\partial \:}{\partial \:x}\left(4+e^{0.5\ln \left(x\right)}+e^{0.25\ln \left(y\right)}\right)=\frac{0.5}{\sqrt{x}}\\ \frac{0.5}{\sqrt{x}}= \frac{0.5}{\sqrt{2.50}}=0.32\\ \frac{\partial \:}{\partial \:y}\left(4+e^{0.5\ln \left(x\right)}+e^{0.25\ln \left(y\right)}\right)=\frac{0.25}{y^{0.75}}=\frac{0.25}{4^{0.75}}=0.088$

Part b

At the solution of the issue, the value of the Lagrange multiplier equals the rate of change in the maximal value of the objective function as the constraint is relaxed.